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I was playing around with variations on the Look-and-say Sequence, and I came up with one that led to consistent stabilization, often resulting in a periodic loop.

Here's how it works:

start with any number

take the first digit of that number

count the total # of occurrences of that digit

remove them from the current number and append the count + digit to a new number in typical look-and-say fashion

move on to the next digit in the current number and repeat until the current number is gone

now repeat the entire process with the new number as the current number, ad infinitum

So, for example:

50

1510 ("one five, one zero")

211510 ("two ones, one five, one zero")

12311510 ("one two, three ones, one five, one zero")

4112131510 ("four ones, one two, one three, one five, one zero")

etc...

if you work out the entire sequence, you get this:

50

1510

211510

12311510

4112131510

145112131510

611425121310

16511422151310

61162514221310

26513215141310

22162551231410

42411625131410

34225116151310

23142225511610

42134114251610

34225113151610

23142225511610

42134114251610

34225113151610

23142225511610
...

Notice that the sequence eventually stabilizes on a three number loop that continues forever. My question is this: is there a name for this Look-and-say variant, and are its properties novel or interesting at all? Is there a simple explanation for why this sequence always stabilizes?

It reminded me of Conway's Game of Life, which is funny since he also did work on the Look-and-say Sequence.

It's neat that you get such small cycles. I wonder if anyone has an explanation for why that's happening. The most I can say is it'll always stabilize because a number with n digits always maps to something with less than 10*ceiling[log(n)+1] digits, and 10*ceiling[log(n)+1] is less than n for sufficiently large n.

This is similar to the Look and Say sequence (method C) listed on oeis: http://oeis.org/A023989
The difference being that the method C digits are re-listed in ascending order. You may be able to submit a Look and Say sequence method D.